Convex Polygons and Other Questions

Date: Mon, 22 Jan 1996 10:35:22 -0500
From: Sweet Home Middle School
Subject: Convex polygons & other questions
Dear Dr. Math:
Our 7th grade advanced math students submitted these questions to
you last week while you were "out". Instead of having them come up
and retype them, I am typing them all in one message and sending them
out to you today for them. I hope this is ok.
Here are their questions:
Given a fixed are, how can I maximze the perimeter of a convex
polygon?
- Mrs. Reimer, Math teacher
How do you find the area of an oval? - Jackie, Robin & Chris
If a convex polygon had 1400 square yards, what would be the maximum
perimeter?
- Zack, Ryan & Jeff
What convex polygon has the greatest perimeter, but has an area of
1400 yds?
- Kristy, Mary, Jill, & Alyssa
What is the largest possible perimeter for an area of 1400 square
yards?
- Nicole, Katie & Kristen
TIA for your help.
Rebecca Silverman
Librarian
Sweet Home Middle School
Amherst, N.Y. 14226 716-837-3500

From: Doctor Jerry
Subject: Re: Convex polygons & other questions
These are interesting questions.
For the first, the answer is that you can't maximize the perimeter,
given a fixed area. To illustrate this, I'll take the area to be 1
and the convex polygon to be an isoceles triangle with base b and
height h.
The area of this triangle is (1/2)*b*h, which we'll set equal to 1.
So,
1=(1/2)*b*h.
The perimeter of this isoceles triangle (just use the Pythagorean
Theorem) is
P = 2*sqrt(h^2+(b/2)^2)+b.
Since h=2/b (from above), the perimeter is
P = 2*sqrt(4/b^2+b^2/4)+b.
If b is taken to be a very small number (which forces the triangle to
be a very narrow but very high triangle), the perimeter becomes large.
If, for example, b = 0.01, then P = 400.01....
If you take b even smaller, the perimeter is even larger.
So, the perimeter can not be maximized.
For the area of an oval, an equation describing the oval would have to
be given. For ovals which are ellipses, there is a known formula.
A circle is a special case of an ellipe. Its area of pi*r^2, where
r is its radius. Ellipses have two radii, as it were, a shortest
radius and a longest radius. If these are b and a, then the area of
an ellipse is pi*a*b. If a = b, then we get the area of a circle.
Questions 3, 4, and 5 appear to be the same as the first question.
There is no maximum. My answer was in terms of triangles. A similar
thing happens for other convex polygons.
-Doctor Jerry, The Math Forum
Check out our web site! http://mathforum.org/dr.math/